Prove2Me

Checking terminal witnesses for a list of stored prefixes

Proved
ThreeSum.LightReportingTerminalFrontier.program_meets

by wurtle · Oct 14, 2026 · Mathlib 0df444a (Lean v4.33.1)

3sumalgorithmsfine-grained-complexitythree-sum-1962word-ram

Write S,Ω,ΔS,\Omega,\Delta for machine space, word magnitude, and depth limits. Fix a global output qq with JJ blocks, a level k≤Jk\le J, and a list of prefixes P0,…,Pm−1P_0,\ldots,P_{m-1}. Let n≤mn\le m, a0,δ∈Za_0,\delta\in\mathbb Z, p∈Np\in\mathbb N, and let L,RL,R be stored integer arrays of lengths NL,NRN_L,N_R. Assume exact source buckets for the residues Li mod pL_i\bmod p and Rj mod p+δR_j\bmod p+\delta modulo the block modulus, exact cover and occurrence data, and the stored left/right masks of these prefixes.

Under the ordered terminal buffers, terminal space and word bounds, bounds on Li,Rj,a0,a0+Li,NL,NRL_i,R_j,a_0,a_0+L_i,N_L,N_R, and depth

d+n+(2h(J−k)h(J−k))+512hJ+10≤Δ,d+n+\binom{2h(J-k)}{h(J-k)}+512hJ+10\le\Delta,

the procedure returns a Boolean rr satisfying

r=1  ⟺  ∃i<n, H(Pi).r=1\iff \exists i<n,\ H(P_i).

Here H(P)H(P) means that an enumerated suffix completion passes both endpoint admissibility tests and finds bucket-matching indices u,vu,v with a0+Lu=Rva_0+L_u=R_v. The stated recursive time bound holds; memory before the terminal mask region is preserved.

This implements terminal witness checking across the supplied prefix list.

Preamble
/-
Copyright (c) 2026 Anthropic, PBC. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
SPDX-License-Identifier: Apache-2.0
-/

import Aesop
import Definitions.Def_APSPSource_RemainingDefinitions
import Definitions.Def_APSPSource_ThreeSumApsp_Lang_Logic
import Definitions.Def_APSPSource_ThreeSumApsp_Lang_Regions
import Definitions.Def_APSPSource_ThreeSumApsp_Lang_Syntax
import Definitions.Def_ThreeSum1962Source_Bundle01
import Definitions.Def_ThreeSum1962Source_Bundle02
import Definitions.Def_ThreeSum1962Source_Bundle04
import Definitions.Def_ThreeSum1962Source_Bundle05
import Definitions.Def_ThreeSum1962Source_Bundle06
import Definitions.Def_ThreeSum1962Source_Bundle07
import Definitions.Def_ThreeSum1962Source_Bundle09
import Definitions.Def_ThreeSum1962Source_Bundle10
import Definitions.Def_ThreeSum1962Source_Bundle11
import Definitions.Def_ThreeSum1962Source_Bundle12
import Definitions.Def_ThreeSum1962Source_Bundle13
import Definitions.Def_ThreeSum1962Source_Bundle14
import Lean.Elab.Tactic.Omega
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Algebra.BigOperators.Group.Finset.Piecewise
import Mathlib.Algebra.BigOperators.Group.Finset.Powerset
import Mathlib.Algebra.BigOperators.Group.Finset.Sigma
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Field.ZMod
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Algebra.Group.Int.Defs
import Mathlib.Algebra.GroupWithZero.Nat
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.BigOperators.Group.List
import Mathlib.Algebra.Order.BigOperators.Group.Multiset
import Mathlib.Algebra.Order.Chebyshev
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Floor.Div
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Algebra.Order.Group.Int
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Algebra.Order.Group.Unbundled.Abs
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.Order.Ring.Pow
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Ring.Nat
import Mathlib.Combinatorics.Colex
import Mathlib.Data.Bool.Basic
import Mathlib.Data.Fin.Embedding
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Finset.Basic
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Max
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Finset.Range
import Mathlib.Data.Finset.Sigma
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Fintype.EquivFin
import Mathlib.Data.Fintype.Fin
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Data.Int.ModEq
import Mathlib.Data.Int.Notation
import Mathlib.Data.Int.SuccPred
import Mathlib.Data.List.Basic
import Mathlib.Data.List.GetD
import Mathlib.Data.List.MinMax
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Pi
import Mathlib.Data.List.ProdSigma
import Mathlib.Data.List.Range
import Mathlib.Data.List.Sublists
import Mathlib.Data.List.TakeDrop
import Mathlib.Data.Nat.Cast.Order.Ring
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Nat.Choose.Bounds
import Mathlib.Data.Nat.Choose.Central
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.Data.Nat.Count
import Mathlib.Data.Nat.Log
import Mathlib.Data.Nat.Notation
import Mathlib.Data.Nat.Size
import Mathlib.Data.Nat.Sqrt
import Mathlib.Data.Nat.SuccPred
import Mathlib.Data.Real.Basic
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Index
import Mathlib.LinearAlgebra.Dual.Lemmas
import Mathlib.LinearAlgebra.Matrix.Defs
import Mathlib.LinearAlgebra.Matrix.Dual
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.Logic.Equiv.Fin.Basic
import Mathlib.Logic.Function.Basic
import Mathlib.NumberTheory.Bertrand
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib.Tactic
import Mathlib.Tactic.Abel
import Mathlib.Tactic.ByContra
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.FinCases
import Mathlib.Tactic.GCongr
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Positivity
import Mathlib.Tactic.Push
import Mathlib.Tactic.Ring
import Mathlib.Tactic.SplitIfs
import Mathlib.Tactic.Tauto
import Theorems.Thm_ThreeSum_BlockSourceCoverInput_Input_of_kept
import Theorems.Thm_ThreeSum_LightReportingTerminal_program_meets


set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option Elab.async false
namespace Light
end Light
namespace ThreeSum.BlockCoefficientRows
end ThreeSum.BlockCoefficientRows
namespace ThreeSum.BlockEncoding
end ThreeSum.BlockEncoding
namespace ThreeSum.BlockForced
end ThreeSum.BlockForced
namespace ThreeSum.BlockIndex
end ThreeSum.BlockIndex
namespace ThreeSum.BlockIndexedQuery
end ThreeSum.BlockIndexedQuery
namespace ThreeSum.BlockRows
end ThreeSum.BlockRows
namespace ThreeSum.BlockSourceEncoding
end ThreeSum.BlockSourceEncoding
namespace ThreeSum.GlobalEncoding
end ThreeSum.GlobalEncoding
namespace ThreeSum.LightBlockQuery
end ThreeSum.LightBlockQuery
namespace ThreeSum.LightBlockSourceBuild
end ThreeSum.LightBlockSourceBuild
namespace ThreeSum.LightHistogram
end ThreeSum.LightHistogram
namespace ThreeSum.LightReportingTerminalMasks
end ThreeSum.LightReportingTerminalMasks
namespace ThreeSum.LightResidues
end ThreeSum.LightResidues
namespace ThreeSum.LightSourceResidues
end ThreeSum.LightSourceResidues
namespace ThreeSum.LocalCoverBounds
end ThreeSum.LocalCoverBounds
namespace ThreeSum.RawIncidences
end ThreeSum.RawIncidences
namespace ThreeSum.RawPrefixCounts
end ThreeSum.RawPrefixCounts
namespace ThreeSum.RawReporting
end ThreeSum.RawReporting
namespace ThreeSum.ReportingHostBucketTest
end ThreeSum.ReportingHostBucketTest
namespace ThreeSum.ReportingHostTerminalFrontier
end ThreeSum.ReportingHostTerminalFrontier
namespace ThreeSum.SourceBuckets
end ThreeSum.SourceBuckets
namespace ThreeSum.StoredReporting
end ThreeSum.StoredReporting
Formal statement
theorem ThreeSum.LightReportingTerminalFrontier.program_meets : ∀ {lim : Light.Limits} {μ : Nat → Int}
  {h J k srcL srcR offsetsL offsetsR flatL flatR NL NR occ maps candidates histL histR inL inR masks masksL masksR base
    bitBase rowOut work n d : Nat}
  (av : Int) (p : Nat) (shift : Int) (left right : Nat → Int) (q : ThreeSum.GlobalEncoding.GlobalOutput h J)
  (pres :
    List.{0}
      (List.{0}
        (Finset.{0}
          (Fin
            (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
              (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h)))))
  (hk : @LE.le.{0} Nat instLENat k J)
  (hn :
    @LE.le.{0} Nat instLENat n
      (@List.length.{0}
        (List.{0}
          (Finset.{0}
            (Fin
              (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h))))
        pres))
  (HL :
    ThreeSum.SourceBuckets.Stored μ
      (fun (i : Nat) =>
        ThreeSum.LightSourceResidues.key p (ThreeSum.BlockSourceEncoding.modulus h J)
          (@OfNat.ofNat.{0} Int (nat_lit 0) (@instOfNat (nat_lit 0))) (left i))
      NL (ThreeSum.BlockSourceEncoding.modulus h J) offsetsL flatL)
  (HR :
    ThreeSum.SourceBuckets.Stored μ
      (fun (i : Nat) => ThreeSum.LightSourceResidues.key p (ThreeSum.BlockSourceEncoding.modulus h J) shift (right i))
      NR (ThreeSum.BlockSourceEncoding.modulus h J) offsetsR flatR)
  (hw : @LE.le.{0} Int Int.instLEInt (@Nat.cast.{0} Int instNatCastInt lim.space) lim.word)
  (hd :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
        (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
          (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
            (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) d n)
            (ThreeSum.LightReportingTerminalMasks.count h J k))
          (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) J
            (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
              (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h)))
        (@OfNat.ofNat.{0} Nat (nat_lit 10) (instOfNatNat (nat_lit 10))))
      lim.depth)
  (hsrcL : @LE.le.{0} Nat instLENat (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) srcL NL) masks)
  (hsrcR : @LE.le.{0} Nat instLENat (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) srcR NR) masks)
  (hoffsetsL :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
        (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) offsetsL
          (ThreeSum.BlockSourceEncoding.modulus h J))
        (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
      masks)
  (hoffsetsR :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)
        (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) offsetsR
          (ThreeSum.BlockSourceEncoding.modulus h J))
        (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
      masks)
  (hflatL : @LE.le.{0} Nat instLENat (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) flatL NL) masks)
  (hflatR : @LE.le.{0} Nat instLENat (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) flatR NR) masks)
  (hocc : @LE.le.{0} Nat instLENat (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) occ J) masks)
  (hmaps :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) maps
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
          (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
            (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))
            (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
              (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h))
          (ThreeSum.GlobalEncoding.scheduleSize h)))
      masks)
  (hcandidates :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) candidates
        (ThreeSum.GlobalEncoding.scheduleSize h))
      masks)
  (hleft :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) histL
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) (ThreeSum.LightLocalCoverSchedule.rows h)
          (ThreeSum.LocalCoverBounds.localPrime h)))
      masks)
  (hright :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) histR
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) (ThreeSum.LightLocalCoverSchedule.rows h)
          (ThreeSum.LocalCoverBounds.localPrime h)))
      masks)
  (hinL :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) inL
        (@List.length.{0}
          (List.{0}
            (Finset.{0}
              (Fin
                (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                  (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h))))
          pres))
      masks)
  (hinR :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) inR
        (@List.length.{0}
          (List.{0}
            (Finset.{0}
              (Fin
                (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                  (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h))))
          pres))
      masks)
  (hmasks :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) masks
        (ThreeSum.LightReportingTerminalMasks.count h J k))
      masksL)
  (hmasksL :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) masksL
        (ThreeSum.LightReportingTerminalMasks.count h J k))
      masksR)
  (hmasksR :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) masksR
        (ThreeSum.LightReportingTerminalMasks.count h J k))
      base)
  (hstop :
    @LE.le.{0} Nat instLENat
      (ThreeSum.LightReportingTerminalPreparation.stop base
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) J
          (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
            (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h))
        (ThreeSum.LightReportingTerminalMasks.count h J k))
      bitBase)
  (hbits :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) bitBase
        (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) J
          (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
            (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h)))
      rowOut)
  (hrowOut :
    @LE.le.{0} Nat instLENat
      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) rowOut
        (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
      work)
  (hspace : @LE.le.{0} Nat instLENat (ThreeSum.LightReportingTerminal.space work h J) lim.space)
  (hword :
    @LE.le.{0} Int Int.instLEInt (@Nat.cast.{0} Int instNatCastInt (ThreeSum.LightReportingTerminal.word h J)) lim.word)
  (hoccRead :
    ∀ (j : Nat),
      @LT.lt.{0} Nat instLTNat j J →
        @Eq.{1} Int (μ (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) occ j))
          (@Nat.cast.{0} Int instNatCastInt
            (@ThreeSum.BlockCoefficientRows.occurrenceIndices h J (@ThreeSum.BlockEncoding.outputOccurrences h J q) j)))
  (hcover : ThreeSum.BlockSourceCoverInput.Input μ h maps candidates histL histR)
  (hreadL :
    ∀ (i : Nat),
      @LT.lt.{0} Nat instLTNat i NL →
        @Eq.{1} Int (μ (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) srcL i)) (left i))
  (hreadR :
    ∀ (j : Nat),
      @LT.lt.{0} Nat instLTNat j NR →
        @Eq.{1} Int (μ (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) srcR j)) (right j))
  (hreadInL :
    ∀ (i : Nat),
      @LT.lt.{0} Nat instLTNat i
          (@List.length.{0}
            (List.{0}
              (Finset.{0}
                (Fin
                  (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                    (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h))))
            pres) →
        @Eq.{1} Int (μ (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) inL i))
          (@Nat.cast.{0} Int instNatCastInt
            (@ThreeSum.SubsetAddress.subsetMask
              (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) J
                (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                  (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h))
              (@ThreeSum.BlockIndexedQuery.leftPrefix J
                (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                  (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h)
                k
                (@ThreeSum.StoredReporting.prefixRows J
                  (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                    (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h)
                  (@ThreeSum.ReportingHostTerminalFrontier.prefixAt h pres i))))))
  (hreadInR :
    ∀ (i : Nat),
      @LT.lt.{0} Nat instLTNat i
          (@List.length.{0}
            (List.{0}
              (Finset.{0}
                (Fin
                  (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                    (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h))))
            pres) →
        @Eq.{1} Int (μ (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) inR i))
          (@Nat.cast.{0} Int instNatCastInt
            (@ThreeSum.SubsetAddress.subsetMask
              (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) J
                (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                  (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h))
              (@ThreeSum.BlockIndexedQuery.rightPrefix J
                (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                  (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h)
                k (@ThreeSum.BlockForced.outputRows h J q)
                (@ThreeSum.StoredReporting.prefixRows J
                  (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                    (@OfNat.ofNat.{0} Nat (nat_lit 512) (instOfNatNat (nat_lit 512))) h)
                  (@ThreeSum.ReportingHostTerminalFrontier.prefixAt h pres i))))))
  (hvaluesL :
    ∀ (i : Nat),
      @LT.lt.{0} Nat instLTNat i NL →
        @LE.le.{0} Int Int.instLEInt (@abs.{0} Int instLatticeInt Int.instAddGroup (left i)) lim.word)
  (hvaluesR :
    ∀ (j : Nat),
      @LT.lt.{0} Nat instLTNat j NR →
        @LE.le.{0} Int Int.instLEInt (@abs.{0} Int instLatticeInt Int.instAddGroup (right j)) lim.word)
  (ha : @LE.le.{0} Int Int.instLEInt (@abs.{0} Int instLatticeInt Int.instAddGroup av) lim.word)
  (htarget :
    ∀ (i : Nat),
      @LT.lt.{0} Nat instLTNat i NL →
        @LE.le.{0} Int Int.instLEInt
          (@abs.{0} Int instLatticeInt Int.instAddGroup
            (@HAdd.hAdd.{0, 0, 0} Int Int Int (@instHAdd.{0} Int Int.instAdd) av (left i)))
          lim.word)
  (hcountWord :
    @LE.le.{0} Int Int.instLEInt
      (@Nat.cast.{0} Int instNatCastInt (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) NL NR))
      lim.word),
  Light.Meets lim ThreeSum.LightReportingTerminalFrontier.program
    (@OfNat.ofNat.{0} Nat (nat_lit 39) (instOfNatNat (nat_lit 39))) d
    (@List.cons.{0} Int av
      (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt srcL)
        (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt srcR)
          (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt offsetsL)
            (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt flatL)
              (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt offsetsR)
                (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt flatR)
                  (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt NL)
                    (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt NR)
                      (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt (ThreeSum.BlockSourceEncoding.modulus h J))
                        (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt J)
                          (@List.cons.{0} Int
                            (@HMul.hMul.{0, 0, 0} Int Int Int (@instHMul.{0} Int Int.instMul)
                              (@OfNat.ofNat.{0} Int (nat_lit 512) (@instOfNat (nat_lit 512)))
                              (@Nat.cast.{0} Int instNatCastInt h))
                            (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt h)
                              (@List.cons.{0} Int
                                (@Nat.cast.{0} Int instNatCastInt (ThreeSum.LocalCoverBounds.localPrime h))
                                (@List.cons.{0} Int
                                  (@Nat.cast.{0} Int instNatCastInt (ThreeSum.LocalCoverBounds.localBucketCap h))
                                  (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt occ)
                                    (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt maps)
                                      (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt candidates)
                                        (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt histL)
                                          (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt histR)
                                            (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt k)
                                              (@List.cons.{0} Int
                                                (@Nat.cast.{0} Int instNatCastInt
                                                  (@ThreeSum.SubsetAddress.subsetMask
                                                    (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) J
                                                      (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                                                        (@OfNat.ofNat.{0} Nat (nat_lit 512)
                                                          (instOfNatNat (nat_lit 512)))
                                                        h))
                                                    (@ThreeSum.BlockIndexedQuery.querySet J
                                                      (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)
                                                        (@OfNat.ofNat.{0} Nat (nat_lit 512)
                                                          (instOfNatNat (nat_lit 512)))
                                                        h)
                                                      k (@ThreeSum.BlockForced.outputRows h J q))))
                                                (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt inL)
                                                  (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt inR)
                                                    (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt masks)
                                                      (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt masksL)
                                                        (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt masksR)
                                                          (@List.cons.{0} Int (@Nat.cast.{0} Int instNatCastInt base)
                                                            (@List.cons.{0} Int
                                                              (@Nat.cast.{0} Int instNatCastInt bitBase)
                                                              (@List.cons.{0} Int
                                                                (@Nat.cast.{0} Int instNatCastInt rowOut)
                                                                (@List.cons.{0} Int
                                                                  (@Nat.cast.{0} Int instNatCastInt work)
                                                                  (@List.cons.{0} Int
                                                                    (@Nat.cast.{0} Int instNatCastInt n)
                                                                    (@List.nil.{0} Int)))))))))))))))))))))))))))))))))
    μ (@ThreeSum.ReportingHostTerminalFrontier.time h J p shift left right NL NR q k pres n)
    fun (r : Int) (ν : Nat → Int) =>
    And
      (Or (@Eq.{1} Int r (@OfNat.ofNat.{0} Int (nat_lit 0) (@instOfNat (nat_lit 0))))
        (@Eq.{1} Int r (@OfNat.ofNat.{0} Int (nat_lit 1) (@instOfNat (nat_lit 1)))))
      (And
        (Iff (@Eq.{1} Int r (@OfNat.ofNat.{0} Int (nat_lit 1) (@instOfNat (nat_lit 1))))
          (@ThreeSum.ReportingHostTerminalFrontier.Hit h J av p shift left right NL NR q k pres n))
        (Light.Kept μ ν masks)) := by
  sorry
Source
https://github.com/c0ldzy17/three-sum-lean/blob/2b639655e07421b9ac743de9090459bab473a7c0/Submission/ThreeSum/LightReportingTerminalFrontier.lean#L40-L151

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